Question

Hi, I was thinking about the next general integral transform, and would like to know what is already known on this integral transform.

The integral transform is:

$$F_{g(x,t)} (f(x)) = \int_{\mathbb{R}} f(x)g(x,t)dx$$

Now as of yet I don't restrict my $g(x,t)$, but I found some rudimentary information on this transform. For example when f is bounded and $g\in L_1(\mathbb{R}^2)$ and $f$ is bounded then $F_g \in L_1(\mathbb{R}^2)$.

I can keep analayze this transform, but if this is already been discussed before in the literature then I would like to read it and not repeating what others have done already.

Thanks in advance.

Answer 1

See Halmos, P.R. and Sunder, V.S. Bounded integral operators on $L^2$ spaces. Springer, Berlin, 1978.